1. Field of the Invention
This invention relates to initial orbit determination (IOD) from angles-only observations. The basis of the methodology is to search a grid of possible boundary values on the range-to-object over the observation interval to find the grid point (by solution of a boundary-value problem such as Lambert's Problem) and initial orbit (orbital element set) that best fits all of the three or more observations according to an error metric.
2. Description of the Related Art
Determining an object's orbit about an astronomical body (e.g. the Earth, Sun, a minor planet, asteroid, etc.) from three line-of-sight (LOS) observations is one of the classical problems of orbit dynamics, being significantly more difficult than solving either of Kepler's equations or Lambert's problem. It resembles Lambert's problem (the derivation of a Keplerian orbit from positions at two given times) in its minimal nature, the difference being that we have two items of data (angles only) at three times instead of three items of data (angle and range) at two times. The greater complexity arises because details of the observers' locations must be available in the angles-only problem, whereas they are entirely irrelevant in Lambert's problem since absolute positions are known. A full characterization of this classic problem and the known solutions are provided by. R. H. Gooding “A New Procedure for Orbit Determination Based on Three Lines of Sight (Angles Only)” Defense Research Agency, 1993, which is excerpted herein and incorporated by reference.
To present the problem in its starkest form, we suppose that nothing is known about the hypothetical orbit, other than through the three observations, except the observed object can be assumed to have a Keplerian motion about a given astronomical body (force centre) having a gravitational strength. Thus all possible solutions are sought, the problem being usually described as one of ‘initial’ orbit determination or IOD. If an approximate orbit can be established from the minimal data available, then a more accurate orbit can be derived later, using as many observations as desired, but that would be an entirely different problem in which the techniques of linearization (relative to the initial orbit), least squares and differential correction can be applied. These techniques are irrelevant until a sufficiently accurate set or orbital parameters is available for differential correction to converge.
Classical solutions to the problem were devised by Laplace and Gauss some two centuries ago. These solutions assumed that orbits would be restricted to certain types and always heliocentric; only a relatively short arc of the orbit would be observed (e.g. a few degrees); observations would be effectively from a single site; and computing methods (e.g. a human) would be severely limited. The human computer did possess one advantage in that the human could improvise as necessary.
In more recent times, solutions that provide greater universality and robustness have been required to handle a wider variety of orbits, a wider range of observed arcs (a few degrees to multiple half-orbits), observations from multiple sites and more timely solutions. A modern restatement of the problem is to find a procedure that is capable of locating all solutions of an arbitrary ‘general problem’; solutions should be determined in the shortest possible time, and to the maximum possible accuracy inherent in the data; intrinsic limitations, associated with some form of indeterminancy, should be recognizable; and there should be options to increase efficiency by the avoidance of solutions that are incompatible with any legitimate assumptions about the orbit. These solutions would have to be built into software of a computer to provide the universality, robust performance and timeliness required.
These factors dictated approaches based on the iteration of the estimated values of the ranges to the object, so called “range iteration” techniques. The current industry standard is the Gooding Initial Orbit Determination (IOD) procedure detailed in the 1993 Defense Research Agency report. The Gooding IOD procedure is part of the Orbit Determination Tool Kit (ODTK), a software package that can be added to the System Took Kit (STK). The ODTK combines both Gooding's IOD tool to determine an initial orbit and a Least Squares differential correction tool to improvide the initial orbit determination based on new angle-only observations.
Gooding's IOD procedure is based on a higher-order Newton correction of the assumed values for two of the three unknown ranges with a Lambert problem solution algorithm (such as Gooding's own solution to the Lambert Problem) at the heart of the procedure. Gooding estimates values for the first and third unknown ranges and solves Lambert's problem to determine a state vector for an initial orbit. The state vector is converted into a universal orbital element set such as (a, e, i, Ω, ω, ν), where a is the semi-major axis of the ellipse, e=SQRT(1−b2/a2) where b is the semi-minor axis is its eccentricity, and the four angles i (inclination), Ω (longitude of ascending node), ω (argument of perigee), and ν (true anomaly) uniquely define the orientation of the orbit. Parameters α (α=μ/a where μ is the gravitational parameter of the Earth) and perigee range q=a*(1−e) may be substituted for a and e. The orbit is propagated to estimate target locations for the second observation. A difference angle between the measured and estimated observation direction is computed. A range step size is determined and used to adjust the first and third estimated ranges. Derivatives of the difference angle are computed with respect to the estimated ranges. Either Halley's method or Newton-Raphson methods of iteration are used to find a local maximum in a search plane. The step-size is reduced and process of computing the derivatives and iterating is repeated until convergence is achieved or the algorithm fails to converge.
Gooding's IOD procedure was specifically configured for the case in which the satellite had completed an unknown number of half-orbits tracing an arc of 180 degrees or more. In these cases Gooding's IOD procedures converges rapidly to a local maximum in approximately 99.9% of the cases. However, when applied to cases in which the satellite traces a small arc of only a few degrees Gooding's IOD procedure may converge in less than approximately 60% of the cases. Approximately 40% of the time Gooding's procedure either failed to converge or converged to a non-physical orbit (e.g. one that escaped Earth orbit or one that intersected Earth orbit).